On the divergence of Hermite-Fejér type interpolation with equidistant nodes

Abstract

If f(x) is defined on [−1, 1], let H¯1 n(f, x) denote the Lagrange interpolation polynomial of degree n (or less) for f which agrees with f at the n+1 equally spaced points xk, n = −1 + (2k)/n (0 ≤ k ≤ n). A famous example due to S. Bernstein shows that even for the simple function h(x) = │x│, the sequence H¯1 n (h, x) diverges as n → ∞ for each x in 0 < │x│ < 1. A generalisation of Lagrange interpolation is the Hermite-Fejér interpolation polynomial H¯mn (f, x), which is the unique polynomial of degree no greater than m(n + 1) – 1 which satisfies (f, Xk, n) = δo, pf(xk, n) (0 ≤ p ≤ m − 1, 0 ≤ k ≤ n). In general terms, if m is an even number, the polynomials H¯mn(f, x) seem to possess better convergence properties than the H¯1 n (f, x). Nevertheless, D.L. Berman was able to show that for g(x) ≡ x, the sequence H¯2n(g, x) diverges as n → ∞ for each x in 0 < │x│. In this paper we extend Berman's result by showing that for any even m, H¯mn(g, x) diverges as n → ∞ for each x in 0 < │x│ < 1. Further, we are able to obtain an estimate for the error │H¯mn(g, x) – g(x)│.

On Lagrange interpolation with equally spaced nodes

“…There is a wide range of literature around Bernstein's classical divergence result. For example, see [2,3,7,9,10,11,12,14]. An extension of Bernstein's result is given in [10]: [2] Theorem 1 informs us that the divergence behaviour is rather general and does not depend on the special characteristics of \x\.…”

“…n-»oo n+ 1 2 For further references, see also Li and Mohapatra [5]. An extension of Theorem 2 to Hermite-Fejer (HF) interpolation with equidistant nodes (but for a different /) is given in Mills and Smith [7]. The aim of this paper is to show that Theorem 2, which is concerned with the rate of divergence of Lagrange interpolation for \x\, is not an isolated phenomenon and thus can be extended to the following result.…”